Let $X$ be a normal variety and $D \subset X$ be a prime divisor which is also normal. It is well-known that we can take a resolution $f: W \to X$ of $X$ such that $$Supp(Exc(f)) \cup \tilde D$$ has simple normal crossings (here $Exc(f)$ is the exceptional locus $f$ and $\tilde D$ is the strict transform of $D$).

My question: can we take $f$ such that the induced morphism $f_D: \tilde D \to D$ is also a log resolution (i.e. $Supp(Exc(f_D))$ has simple normal crossings)?

Let $X$ be a normal variety and $D \subset X$ be a prime divisor which is also normal. It is well-known that we can take a resolution $f: W \to X$ of $X$ such that $$\DeclareMathOperator{\Supp}{\operatorname{Supp}} \DeclareMathOperator{\Exc}{\operatorname{Exc}} \Supp(\Exc(f)) \cup \tilde D $$ has simple normal crossings (here $\Exc(f)$ is the exceptional locus $f$ and $\tilde D$ is the strict transform of $D$).

My question: can we take $f$ such that the induced morphism $f_D: \tilde D \to D$ is also a log resolution (i.e. $\Supp(\Exc(f_D))$ has simple normal crossings)?

Let $X$ be a normal variety and $D \subset X$ be a prime divisor which is also normal. It is well-known that we can take a resolution of $f: W \to X$ such that $$Supp(Exc(f)) \cup \tilde D$$ has simple normal crossings (here $Exc(f)$ is the exceptional locus $f$ and $\tilde D$ is the strict transform of $D$).

My questions: can we take $f$ such that the induced morphism $f_D: \tilde D \to D$ is also a log resolution (i.e. $Supp(Exc(f_D))$ has simple normal crossings)?

Let $X$ be a normal variety and $D \subset X$ be a prime divisor which is also normal. It is well-known that we can take a resolution $f: W \to X$ of $X$ such that $$Supp(Exc(f)) \cup \tilde D$$ has simple normal crossings (here $Exc(f)$ is the exceptional locus $f$ and $\tilde D$ is the strict transform of $D$).

My question: can we take $f$ such that the induced morphism $f_D: \tilde D \to D$ is also a log resolution (i.e. $Supp(Exc(f_D))$ has simple normal crossings)?